The problem of finding covariance matrices that remain constant in time for arbitrary multi-dimensional quadratic Hamiltonians (including those with time-dependent coefficients) is considered. General solutions are obtained.
Diagonalisation of quadratic multi-mode, time-dependent Hamiltonians can be fraught with difficulties. We set about diagonalising Hamiltonians from quantum optics. The Hamiltonians we choose represent Raman scattering, parametric amplification and the frequency converter. We present a systematic approach to this problem. The process of diagonalisation allows us to find invariants of the Hamiltonians under time evolution, find ground-state energies, apply the theory of Greens functions to find the solution to Schrodinger's equation, examine the conditions for oscillation or growth, and squeezing.